Cross-connections and variants of the full transformation semigroup

P. A. Azeef Muhammed

Acta Sci. Math. (Szeged) 84:3-4(2018), 377-399
44/2017

Abstract. Cross-connection theory propounded by Nambooripad describes the ideal structure of a regular semigroup using the categories of principal left (right) ideals. A variant $\mathscr{T}_X^\theta $ of the full transformation semigroup $(\mathscr{T}_X,\cdot )$ for an arbitrary $\theta\in \mathscr{T}_X$ is the semigroup $\mathscr{T}_X^\theta = (\mathscr{T}_X,\ast )$ with the binary operation $\alpha\ast \beta = \alpha\cdot \theta\cdot \beta $ where $\alpha, \beta\in \mathscr{T}_X$. In this article, we describe the ideal structure of the regular part ${\msbm R}eg (\mathscr{T}_X^\theta )$ of the variant of the full transformation semigroup using cross-connections. We characterize the constituent categories of ${\msbm R}eg (\mathscr{T}_X^\theta )$ and describe how they are \emph{cross-connected} by a functor induced by the sandwich transformation $\theta $. This leads us to a structure theorem for the semigroup and gives the representation of ${\msbm R}eg (\mathscr{T}_X^\theta )$ as a cross-connection semigroup. Using this, we give a description of the biordered set and the sandwich sets of the semigroup.